Our motivation in this paper comes from a beautiful work of Huneke and Wiegand [1]}, which reproves and extends the aforementioned rigidity result of Luk and Yau via the machinery of commutative algebra. Huneke and Wiegand investigates suitable conditions under which if one local cohomology module of a tensor product of finitely generated modules vanishes, then all lower ones vanish. As a consequence, Huneke and Wiegand [1]} obtained remarkable results that relate Serre's conditions to the vanishing of a single local cohomology module of a tensor product.
